A simple algebraic proof of the equational interpolation theorem
(1991)

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"... It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which ..."

It is well known that interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications, they have also many other useful logical consequences. In this paper, we solve the interpolation problem for Grothendieck institutions which have recently emerged as an important mathematical structure underlying heterogenous multi-logic specification. Our main result can be used in the applications in several different ways. It can be used to establish interpolation properties for multi-logic Grothendieck institutions, but also to lift interpolation properties from unsorted logics to their many sorted variants. The importance of the latter resides in the fact that, unlike other structural properties of logics, many sorted interpolation is a non-trivial generalisation of unsorted interpolation. The concepts, results, and the applications discussed in this paper are illustrated with several examples from conventional logic and algebraic specification theory.

"... theory, categorical logic. model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms repre ..."

theory, categorical logic. model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum

"... : Although modularisation is basic to modern computing, it has been little studied for logic-based programming. We treat modularisation for equational logic programming using the institution of category-based equational logic in three different ways: (1) to provide a generic satisfaction conditio ..."

: Although modularisation is basic to modern computing, it has been little studied for logic-based programming. We treat modularisation for equational logic programming using the institution of category-based equational logic in three different ways: (1) to provide a generic satisfaction condition for equational logics; (2) to give a category-based semantics for queries and their solutions; and (3) as an abstract definition of compilation from one (equational) logic programming language to another. Regarding (2), we study soundness and completeness for equational logic programming queries and their solutions. This can be understood as ordinary soundness and completeness in a suitable &quot;non-logical&quot; institution. Soundness holds for all module imports, but completeness only holds for conservative module imports. Categorybased equational signatures are seen as modules, and morphisms of such signatures as module imports. Regarding (3), completeness corresponds to compiler correc...

"... Abstract. Interpolation results are investigated for various types of for-mulae. By shifting the focus from syntactic to semantic interpolation, we generate, prove and classify more than twenty interpolation results for first-order logic and some for richer logics. A few of these results non-trivial ..."

Abstract. Interpolation results are investigated for various types of for-mulae. By shifting the focus from syntactic to semantic interpolation, we generate, prove and classify more than twenty interpolation results for first-order logic and some for richer logics. A few of these results non-trivially generalize known interpolation results. All the others are new. 1

"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the so-caled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a ..."

We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the so-caled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.

by
Razvan Diaconescu
- UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE, 2008

"... ... model theoretic framework of the theory of institutions. For this semantics we develop a generic interpolation result which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of i ..."

... model theoretic framework of the theory of institutions. For this semantics we develop a generic interpolation result which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of important applications to computing science especially in the area of structured specifications.

"... In this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions, which means two levels of institution independence. This abstract axiomatic app ..."

In this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions, which means two levels of institution independence. This abstract axiomatic approach provides a uniform framework for the study of structured specifications independently from any actual choice of specification building operators, and moreover it unifies the theory and the model oriented approaches. Within this framework we develop concepts and results about ‘abstract structured specifications ’ such as co-limits, model amalgamation, compactness, interpolation, sound and complete proof theory, and pushout-style parameterization with sharing, all of them in a top down manner dictated by the upper level of institution independence. 1.

"... : We introduce the concept of semantic paramodulation as a "semantic " definition of paramodulation (in the sense that it applies to any model, not only to the term algebra) within the framework of category-based equational logic (introduced by [8, 9]). This not only generalises the tradit ..."

: We introduce the concept of semantic paramodulation as a &quot;semantic &quot; definition of paramodulation (in the sense that it applies to any model, not only to the term algebra) within the framework of category-based equational logic (introduced by [8, 9]). This not only generalises the traditional syntactic approaches to paramodulation, but also provides an abstract framework for a unified treatment of paramodulation and its refinements in a wide class of equational logical systems. This includes many sorted algebra, order sorted algebra, rewrite models, Horn clause logic, equational logic modulo axioms, constraint logic, and any combination of these. The possibility to give a direct treatment of refutations modulo axioms and to constraint computations is of special importance. Moreover, semantic paramodulation gives the possibility to develop equational logic programming over non-conventional structures (such as graphs) whose models might be essentially different from the traditional set...